Problem: Simplify and expand the following expression: $ \dfrac{3}{k - 7}+ \dfrac{2}{2k - 20}- \dfrac{4k}{k^2 - 17k + 70} $
Explanation: First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor a $2$ out of denominator in the second term: $ \dfrac{2}{2k - 20} = \dfrac{2}{2(k - 10)}$ We can factor the quadratic in the third term: $ \dfrac{4k}{k^2 - 17k + 70} = \dfrac{4k}{(k - 7)(k - 10)}$ Now we have: $ \dfrac{3}{k - 7}+ \dfrac{2}{2(k - 10)}- \dfrac{4k}{(k - 7)(k - 10)} $ The least common multiple of the denominators is: $ (k - 7)(k - 10)$ In order to get the first term over $(k - 7)(k - 10)$ , multiply by $\dfrac{2(k - 10)}{2(k - 10)}$ $ \dfrac{3}{k - 7} \times \dfrac{2(k - 10)}{2(k - 10)} = \dfrac{6(k - 10)}{(k - 7)(k - 10)} $ In order to get the second term over $(k - 7)(k - 10)$ , multiply by $\dfrac{k - 7}{k - 7}$ $ \dfrac{2}{2(k - 10)} \times \dfrac{k - 7}{k - 7} = \dfrac{2(k - 7)}{(k - 7)(k - 10)} $ In order to get the third term over $(k - 7)(k - 10)$ , multiply by $\dfrac{2}{2}$ $ \dfrac{4k}{(k - 7)(k - 10)} \times \dfrac{2}{2} = \dfrac{8k}{(k - 7)(k - 10)} $ Now we have: $ \dfrac{6(k - 10)}{(k - 7)(k - 10)} + \dfrac{2(k - 7)}{(k - 7)(k - 10)} - \dfrac{8k}{(k - 7)(k - 10)} $ $ = \dfrac{ 6(k - 10) + 2(k - 7) - 8k} {(k - 7)(k - 10)} $ Expand: $ = \dfrac{6k - 60 + 2k - 14 - 8k}{2k^2 - 34k + 140} $ $ = \dfrac{-74}{2k^2 - 34k + 140}$ Simplify: $ = \dfrac{-37}{k^2 - 17k + 70}$